A Selection Theorem for the Carathéodory Kernel Convergence of Pointed Domains

Abstract We present a selection theorem for domains in $$\mathbb C^n$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msup> <mml:mi>C</mml:mi> <mml:mi>n</mml:mi> </mml:msup> </mml:math> , $$n\ge 1$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , which states that any tamed sequence of pointed connected open subsets admits a subsequence convergent to its own kernel in the sense of Carathéodory. Not only is this analogous to the well-known Blaschke selection theorem for compact convex sets, but it fits better in the study of normal families of biholomorphic maps with varying domains and ranges.